Starting from a special type of function on a finite dimensional group called a cusp form, we define an object on the corresponding loop group which depends on one complex variable. This object, the cuspidal loop Eisenstein series, can then be shown to be entire on the complex plane, which is a phenomenon quite unusual from the point of view of finite-dimensional automorphic forms. We explain how to deduce this result from two ingredients: (a) inequalities between the classical and central "directions" of elements in a certain discrete family in a loop symmetric space; and (b) a strengthening of the usual rapid-decay statements for cusp forms on finite-dimensional groups.

This is joint work with H. Garland and S.D. Miller